Continuous functions are, in essence, functions whose graphs can be drawn without lifting up your pen. This may sound simple, but this is in fact a very rich subject. Learn how continuity is defined using limits, and about a main property of all continuous functions -- the Intermediate value theorem.
Differentiating functions is not an easy task! Make your first steps in this vast and rich world with some of the most basic differentiation rules, including the Power rule. It will surely make you feel more powerful.
The chain rule sets the stage for implicit differentiation, which in turn allows us to differentiate inverse functions (and specifically the inverse trigonometric functions). This is really the top of the line when it comes to differentiation.
Limits are intuitive, yet elusive. Learn what they are all about and how to find limits of functions from graphs or tables of values. Learn about the difference between one-sided and two-sided limits and how they relate to each other.
Analyzing_functions_with_calculus
Let's put all of our differentiation abilities to use, by analyzing the graphs of various functions. As you will see, the derivative and the second derivative of a function can tell us a lot about the function's graph.
Now that we have all the conceptual stuff laid down, we can start have some fun with finding limits of various functions. Some of these limits don't want you to find them so fast, but we're sure you'll get them in the end!
Basically, a limit must be at a specific point and have a specific value in order to be defined. Nevertheless, there are two kinds of limits that break these rules. One kind is unbounded limits -- limits that approach ± infinity (you may know them as "vertical asymptotes"). The other kind is limits at infinity -- these limits describe the value a function is approaching as x goes to ± infinity (you may know them as "horizontal asymptotes").
Differentiating_common_functions
Now that you know all the important differentiation rules, let's solve some problems that involve the differentiation of various common functions.
Solve real world problems (and some pretty elaborate mathematical problems) using the power of differential calculus.
Covered basic differentiation? Great! Now let's take things to the next level. In this topic, you will learn general rules that tell us how to differentiate products of functions, quotients of functions, and composite functions. Anxious to find the derivative of eˣ⋅sin(x²)? You've come to the right place.
Get comfortable with the big idea of differential calculus, the derivative. The derivative of a function has many different interpretations and they are all very useful when dealing with differential calculus problems. This topic covers all of those interpretations, including the formal definition of the derivative and the notion of differentiable functions.
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